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In units $c=1$, a slowly moving isolated source has trace-free mass quadrupole moment $Q^{ij}=\int T^{00}(x^i{x}^j-{\delta }^{ij}|\mathbf{x}{|}^2/3)d^{3}x$. The retarded linearized gravity solution and the stress-energy quadrupole identity give the displayed physical spatial strain, with $\mathbf{n}=\mathbf{x}/r$ and the transverse-traceless projector $\Lambda$. If the spatial metric is written $g_{ij}=-{\delta }_{ij}-2E_{ij}$ for metric signature $(+---)$, its trace-free Lorenz-gauge coefficient before the transverse projection is $E_{ij}=G{\ddot{Q}}_{ij}(t-r)/r$. Thus $H^{\mathrm{TT}}=2E^{\mathrm{TT}}$. Far distance alone does not permit dropping source-dependent retardation: the quadrupole approximation also requires the source size to be small compared with the radiation wavelength.

For a weakly gravitating thin spherical shell of mass $M$, radius $R$ and uniform angular velocity $\Omega$, its interior linearized gravity metric has constant scalar potentials $A=-GM/R$, $C=GM/R$ and $g_{0i}=\omega (\hat{\mathbf{z}}\times \mathbf{x}{)}_i$. The scalar mass shell is a monopole. Its mass current is a dipole: regular interior and decaying exterior radial solutions are proportional to $r$ and $r^{-2}$, and their derivative jump fixes the factor $4/3$. Local inertial axes co-rotate with the shell at $\omega$ relative to asymptotic nonrotating axes. This is the leading interior Lense-Thirring precession, neglecting quadratic rotation and higher gravitational orders.

Spatial moments of a mass density distribution organize its far gravitational field. Symmetric trace-free parts isolate angular multipoles; the second moment gives the mass quadrupole moment. In slow-motion linearized gravity, changes of the quadrupole produce the leading gravitational radiation; the conserved monopole and centre-of-mass dipole do not radiate independently.

Work consistently to first order in the metric perturbation, with ${\partial }_{\mu }{\xi }^{\nu }$ of the same small order. Under the passive change $x^{\prime \mu }=x^{\mu }+{\xi }^{\mu }(x)$,Applying this to the full metric tensor and writing $g_{\mu \nu }^{\prime }={\eta }_{\mu \nu }+h_{\mu \nu }^{\prime }$ with the same fixed numerical Minkowski metric givesIndices on $\xi$ are lowered with $\eta$. This is the gauge transformation of the perturbation relative to a fixed background. If the background were transformed as a tensor as well, the difference of those two fully transformed metrics would instead obey the homogeneous tensor transformation law. The inhomogeneous terms arise from retaining the fixed-background identification used in linearized gravity.

The trace changes by $h^{\prime }=h-2{\partial }_{\alpha }{\xi }^{\alpha }$, so the trace-reversed metric perturbation changes asLet $F_{\nu }={\partial }^{\mu }{\bar{h}}_{\mu \nu }$. Commuting partial derivatives givesThus choose a solution of the inhomogeneous wave equationSuch a solution exists locally with suitable initial data, and can be selected with appropriate wave-equation boundary conditions. This imposes the Lorenz gauge in linearized gravity, ${\partial }^{\mu }{\bar{h}}_{\mu \nu }^{\prime }=0$. All the divergence terms in the supplied vacuum equation then vanish, leavingThe remaining freedom $\square {\xi }_{\nu }=0$ is the residual gauge symmetry of linearized gravity.

Using the old coordinates in these derivatives is consistent at the retained order. IndeedActing on a first-order perturbation, the correction is a product of two first-order quantities and is discarded. Similarly replacing the argument $x^{\prime }$ by $x$ changes $h^{\prime }$ by $O(h^2)$ under the same perturbative ordering. This does not assert an exact equality of the two coordinate differential operators; it is the linear approximation about the fixed flat background.

Using the source's rounded solar mass, each hole has $m\simeq 3\times 10^{31}\mathrm{kg}$. Its Schwarzschild radius isThus $\boxed{r_s\sim 10^5\mathrm{m}}$ at order-of-magnitude accuracy. Using the more accurate solar mass would give about $9\times 10^4\mathrm{m}$.

The stated orbit is not a consistent separated-hole weak-field binary. Its nominal center-to-center separation is only $20\mathrm{km}$, below even the above single-hole radius. Applying two isolated Schwarzschild spacetime descriptions and a circular Newtonian orbit at that separation is not controlled. Likewise an estimate ${\bar{h}}_{11}^{\prime }\sim r_s/r$ at $r=10\mathrm{km}$ is of order one or larger, contradicting the small-perturbation assumption. The intended near-source statement is therefore a strong-field scale,not a reliable prediction from linearized gravity at the specified orbit. The exact radiative component is also gauge dependent near the source.

One can still estimate the intended distant gravitational wave strain scale. A radiative amplitude decreases as $1/D$ in the wave zone. A compact strong-field source has a characteristic length of order $r_s$, giving a dimensional strain estimate $h_{\mathrm{far}}\sim r_s/D$, up to mass, orientation and efficiency factors of order unity. This is also the strong-field dimensional scale suggested by far-zone quadrupole strain; it is not an application of the slow-motion formula to the inconsistent orbit.

With the supplied conversion,For a ring radius $a=4\times 10^3\mathrm{m}$, part (c) gives a single principal semiaxis displacement amplitudeThe difference of the two principal semiaxes has twice this amplitude. This is a conditional dimensional estimate of a physically plausible merger signal at that distance; the supplied close circular orbit does not define a self-consistent waveform or a precise amplitude.

Gravitational waves can exhibit interference. In linearized gravity, overlapping strains add, so coherent contributions can enhance or cancel the measured response. Independent astrophysical sources generally have unrelated phases and average to a stochastic background rather than stable fringes. More plausible coherent paths arise when gravitational lensing sends different parts of the same signal to the detector with different delays; fringes require sufficient overlap and coherence. The relevant quadratic radiative energy flux contains the cross terms even though the leading wave equation is linear.

Expand the Einstein tensor about the Minkowski metric, keeping a second-order metric perturbation as well:Here $G^{(2)}[h,h]$ includes all quadratic terms from the Einstein tensor, including the inverse-metric corrections in its contractions. In vacuum, the first-order Linearized Einstein equations give $G^{(1)}[h]=0$, while the next order gives $G^{(1)}[j]=-G^{(2)}[h,h]$. Define the quadratic gravitational effective stress-energy tensor byIt is symmetric because the Einstein tensor is symmetric, and is quadratic in $h$ and its derivatives. The linearized contracted Bianchi identity is the off-shell identity ${\partial }^{\mu }{G}_{\mu \nu }^{(1)}[j]=0$. The second-order vacuum equation therefore impliesOne can obtain the same conclusion directly by expanding the full contracted Bianchi identity to second order: every correction to the flat divergence multiplies $G^{(1)}[h]$ and vanishes on the first-order vacuum solution. Thus conservation holds for vacuum solutions of linearized gravity, rather than for an arbitrary off-shell $h$. Relative to the fixed Minkowski background, $t$ is a tensor under background changes of coordinates, but this does not make it invariant under the perturbative gauge freedom.

The Einstein tensor is $G_{\alpha \beta }=R_{\alpha \beta }-\frac{1}{2}{g}_{\alpha \beta }R$. Taking the metric trace of the Einstein field equations in four dimensions givesand substitution back gives the Trace-reversed Einstein field equations in D dimensions specialized to $D=4$:For linearized gravity, the given equation and $G_{\mu \nu }=8\pi T_{\mu \nu }$ implySince $\bar{h}=-h$, its trace is $\square h=16\pi T$. Using ${\bar{h}}_{\mu \nu }=h_{\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }h$ then yields